The influence of large coalitions

This paper contains two results on influence in collective decision games. The first part deals with general perfect information coin-flipping games as defined in [3].Baton passing (see [8]), ann-player game from this class is shown to have the following property: IfS is a coalition of size at most $$\frac{n}{{3\log n}}$$, then the influence ofS on the game is only $$O\left( {\frac{{\left| S \right|}}{n}} \right)$$. This complements a result from [3] that for everyk there is a coalition of sizek with influence O(k/n). Thus the best possible bounds on influences of coalitions of size up to this threshold are known, and there need not be coalitions up to this size whose influence asymptotically exceeds their fraction of the population. This result may be expected to play a role in resolving the most outstanding problem in this area: Does everyn-player perfect information coin flipping game have a coalition ofo(n) players with influence 1-o(1)? (Recently Alon and Naor [1] gave a negative answer to this question.) In a recent paper Kahn, Kalai and Linial [7] showed that for everyn-variable boolean function of expectation bounded away from zero and one, there is a set of $$\frac{{n\omega (n)}}{{\log n}}$$ variables whose influence is 1-o(1), wherew(n) is any function tending to infinity withn. They raised the analogous question where 1-o(1) is replaced by any positive constant and speculated that a constant influence may be always achievable by significantly smaller sets of variables. This problem is almost completely solved in the second part of this article where we establish the existence of boolean functions where only sets of at least $$\Omega \left( {\frac{n}{{\log ^2 n}}} \right)$$ variables can have influence bounded away from zero.